infinitely small

Of course almost any set can be given some kind of group structure, but one which ahs nothing at all to do with its topological structure. E.g. any finite set can be made into a cyclic group and any countable set can be made into the integer group, and any set with cardinality equal to that of the reals can be made into the reals as a group.

mr small, the key geometric property of groups that are also manifolds, is that they have trivial tangent bundle. Inparticular they admit a vector field without zeroes. Hence the euler characteristic must be zero. In the case of surfaces this shows that the genus must be one, the only case of a smooth surfce which also has a (smooth) group structure.